Error bounds theory

Alternatively, one can use error bound theory (see Chapter III) if any kind of explicit termination of the J-fraction (3.3) is avoided. This approach has been followed extensively in the literature by the different schools. The approximate local density of states is obtained by differentiation of the two continuous curves N+ E) and N E), which are upper and lower bounds of the integrated density of states... 

It is, nervertheless, highly desirable, to construct a theory that maintains the simplicity of the BCS theory, yet removes the (major part of the) BCS errors. A theory, recently developed by Li [8], which uses the number-conserving quasi-particle (NCQP) method, is such a theory. We are now working on replacing BCS+BET by NCQP+BET, and on resuming realistic calculations. This time we shall be able to go much beyond the bounds imposed by the BCS approximation. 

In addition to this, it is often important to have also exact mathematical criteria of the truncation errors performed when an infinite continued fraction is replaced by its approximant of order n. In a number of situations these criteria do exist and are also economical. Keeping in line with the general spirit of our review, we focus on error bounds in continued fractions, describing specific (though sufficiently general) physical problems. We do not dwell on the interesting theory of error bounds in more abstract situations. 

Out of the detailed mathematical aspects, some of them summarized in this section, there is a more general physical concept at the heart of the theory of error bounds. It is a fact that the memory function formalism provides in a natural manner a framework by which Ae short-time behavior, via the kernel of the integral equations, makes its effects felt in the long-time tail. The mathematical apparatus of continued fractions can adequately describe memory effects, and this explains the central role of this tool in the theory of relaxation. 

In spite of these useful investigations, further work must still be done to assess the accuracy in realistic situations such as those encountered in band structure calculations on real crystals. Notice, furthermore, that any kind of extrapolation of continued fraction parameters must be consistent with the theory of error bounds as provided by the first few exact available moments. 

A. J. Viterbi (1967) Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Informat. Theory IT-13, pp. 260-269... 

Cl) (perturbation theory (PT)). The PT energy errors are all within the chemical accuracy of 0.043 eV (1 kcal/mol). These error bounds are highly satisfactory and we also found that these error bounds do not increase with the number of MCP... 

Shannon, C.E. 1948. A mathematical theory of communication. Bell Sys. Tech.. 27 379 423, 623 656. Ungerboeck, G. 1982. Channel coding with multilevel/phase signals. IEEE Trans. Inf Theory (Jan.). Viterbi, A.J. 1967. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory IT-13 260-269. 

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